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A [[Probability distribution|probability distribution]] of a random variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066180/n0661801.png" /> which takes non-negative integer values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066180/n0661802.png" /> in accordance with the formula
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066180/n0661803.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
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{{TEX|done}}
  
for any real values of the parameters <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066180/n0661804.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066180/n0661805.png" />. The [[Generating function|generating function]] and the [[Characteristic function|characteristic function]] of a negative binomial distribution are defined by the formulas
+
A [[Probability distribution|probability distribution]] of a random variable  $  X $
 +
which takes non-negative integer values  $  k = 0, 1 \dots $
 +
in accordance with the formula
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066180/n0661806.png" /></td> </tr></table>
+
$$ \tag{* }
 +
{\mathsf P} \{ X = k \}  = \
 +
\left ( \begin{array}{c}
 +
r+ k- 1 \\
 +
k
 +
\end{array}
 +
\right ) p  ^ {r} ( 1- p)  ^ {k}
 +
$$
 +
 
 +
for any real values of the parameters  $  0 < p < 1 $
 +
and  $  r > 0 $.
 +
The [[Generating function|generating function]] and the [[Characteristic function|characteristic function]] of a negative binomial distribution are defined by the formulas
 +
 
 +
$$
 +
P( z)  = p  ^ {r} ( 1- qz)  ^ {-} r
 +
$$
  
 
and
 
and
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066180/n0661807.png" /></td> </tr></table>
+
$$
 +
f( t)  = p  ^ {r} ( 1- qe  ^ {it} )  ^ {-} r ,
 +
$$
 +
 
 +
respectively, where  $  q = 1- p $.
 +
The mathematical expectation and variance are equal, respectively, to  $  rq= p $
 +
and  $  rq/p  ^ {2} $.
 +
The distribution function of a negative binomial distribution for the values  $  k = 0, 1 \dots $
 +
is defined in terms of the values of the [[Beta-distribution|beta-distribution]] function at a point  $  p $
 +
by the following relation:
 +
 
 +
$$
 +
F( k)  =  {\mathsf P} \{ X < k \}  = \
  
respectively, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066180/n0661808.png" />. The mathematical expectation and variance are equal, respectively, to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066180/n0661809.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066180/n06618010.png" />. The distribution function of a negative binomial distribution for the values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066180/n06618011.png" /> is defined in terms of the values of the [[Beta-distribution|beta-distribution]] function at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066180/n06618012.png" /> by the following relation:
+
\frac{1}{B( r, k+ 1) }
 +
\int\limits _ { 0 } ^ { p }  x  ^ {r-} 1 ( 1- x)  ^ {k}
 +
dx,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066180/n06618013.png" /></td> </tr></table>
+
where  $  B( r, k+ 1) $
 +
is the beta-function.
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066180/n06618014.png" /> is the beta-function.
+
The origin of the term  "negative binomial distribution" is explained by the fact that this distribution is generated by a [[Binomial|binomial]] with a negative exponent, i.e. the probabilities (*) are the coefficients of the expansion of  $  p  ^ {r} ( 1- qz)  ^ {-} r $
 +
in powers of  $  z $.
  
The origin of the term  "negative binomial distribution"  is explained by the fact that this distribution is generated by a [[Binomial|binomial]] with a negative exponent, i.e. the probabilities (*) are the coefficients of the expansion of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066180/n06618015.png" /> in powers of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066180/n06618016.png" />.
+
Negative binomial distributions are encountered in many applications of probability theory. For an integer  $  r > 0 $,
 +
the negative binomial distribution is interpreted as the distribution of the number of failures before the  $  r $-
 +
th  "successin a scheme of [[Bernoulli trials|Bernoulli trials]] with probability of  "success"  $  p $;
 +
in this context it is usually called a [[Pascal distribution|Pascal distribution]] and is a discrete analogue of the [[Gamma-distribution|gamma-distribution]]. When  $  r= 1 $,
 +
the negative binomial distribution coincides with the [[Geometric distribution|geometric distribution]]. The negative binomial distribution often appears in problems related to the randomization of the parameters of a distribution; for example, if  $  Y $
 +
is a random variable having, conditionally on  $  \lambda $,
 +
a [[Poisson distribution|Poisson distribution]] with random parameter  $  \lambda $,
 +
which in turn has a gamma-distribution with density
  
Negative binomial distributions are encountered in many applications of probability theory. For an integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066180/n06618017.png" />, the negative binomial distribution is interpreted as the distribution of the number of failures before the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066180/n06618018.png" />-th  "success"  in a scheme of [[Bernoulli trials|Bernoulli trials]] with probability of  "success"  <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066180/n06618019.png" />; in this context it is usually called a [[Pascal distribution|Pascal distribution]] and is a discrete analogue of the [[Gamma-distribution|gamma-distribution]]. When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066180/n06618020.png" />, the negative binomial distribution coincides with the [[Geometric distribution|geometric distribution]]. The negative binomial distribution often appears in problems related to the randomization of the parameters of a distribution; for example, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066180/n06618021.png" /> is a random variable having, conditionally on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066180/n06618022.png" />, a [[Poisson distribution|Poisson distribution]] with random parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066180/n06618023.png" />, which in turn has a gamma-distribution with density
+
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066180/n06618024.png" /></td> </tr></table>
+
\frac{1}{\Gamma ( \mu ) }
 +
x ^ {\mu - 1 } e ^ {- \alpha x } ,\ \
 +
x > 0,\  \mu > 0,
 +
$$
  
then the marginal distribution of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066180/n06618025.png" /> will be a negative binomial distribution with parameters <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066180/n06618026.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066180/n06618027.png" />. The negative binomial distribution serves as a limiting form of a [[Pólya distribution|Pólya distribution]].
+
then the marginal distribution of $  Y $
 +
will be a negative binomial distribution with parameters $  r = \mu $
 +
and $  p = \alpha /( 1+ \alpha ) $.  
 +
The negative binomial distribution serves as a limiting form of a [[Pólya distribution|Pólya distribution]].
  
The sum of independent random variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066180/n06618028.png" /> which have negative binomial distributions with parameters <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066180/n06618029.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066180/n06618030.png" />, respectively, has a negative binomial distribution with parameters <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066180/n06618031.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066180/n06618032.png" />. For large <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066180/n06618033.png" /> and small <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066180/n06618034.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066180/n06618035.png" />, the negative binomial distribution is approximated by the Poisson distribution with parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066180/n06618036.png" />. Many properties of a negative binomial distribution are determined by the fact that it is a generalized Poisson distribution.
+
The sum of independent random variables $  X _ {1} \dots X _ {n} $
 +
which have negative binomial distributions with parameters $  p $
 +
and $  r _ {1} \dots r _ {n} $,  
 +
respectively, has a negative binomial distribution with parameters $  p $
 +
and $  r _ {1} + \dots + r _ {n} $.  
 +
For large $  r $
 +
and small $  q $,  
 +
where $  rq \sim \lambda $,  
 +
the negative binomial distribution is approximated by the Poisson distribution with parameter $  \lambda $.  
 +
Many properties of a negative binomial distribution are determined by the fact that it is a generalized Poisson distribution.
  
 
====References====
 
====References====

Latest revision as of 08:02, 6 June 2020


A probability distribution of a random variable $ X $ which takes non-negative integer values $ k = 0, 1 \dots $ in accordance with the formula

$$ \tag{* } {\mathsf P} \{ X = k \} = \ \left ( \begin{array}{c} r+ k- 1 \\ k \end{array} \right ) p ^ {r} ( 1- p) ^ {k} $$

for any real values of the parameters $ 0 < p < 1 $ and $ r > 0 $. The generating function and the characteristic function of a negative binomial distribution are defined by the formulas

$$ P( z) = p ^ {r} ( 1- qz) ^ {-} r $$

and

$$ f( t) = p ^ {r} ( 1- qe ^ {it} ) ^ {-} r , $$

respectively, where $ q = 1- p $. The mathematical expectation and variance are equal, respectively, to $ rq= p $ and $ rq/p ^ {2} $. The distribution function of a negative binomial distribution for the values $ k = 0, 1 \dots $ is defined in terms of the values of the beta-distribution function at a point $ p $ by the following relation:

$$ F( k) = {\mathsf P} \{ X < k \} = \ \frac{1}{B( r, k+ 1) } \int\limits _ { 0 } ^ { p } x ^ {r-} 1 ( 1- x) ^ {k} dx, $$

where $ B( r, k+ 1) $ is the beta-function.

The origin of the term "negative binomial distribution" is explained by the fact that this distribution is generated by a binomial with a negative exponent, i.e. the probabilities (*) are the coefficients of the expansion of $ p ^ {r} ( 1- qz) ^ {-} r $ in powers of $ z $.

Negative binomial distributions are encountered in many applications of probability theory. For an integer $ r > 0 $, the negative binomial distribution is interpreted as the distribution of the number of failures before the $ r $- th "success" in a scheme of Bernoulli trials with probability of "success" $ p $; in this context it is usually called a Pascal distribution and is a discrete analogue of the gamma-distribution. When $ r= 1 $, the negative binomial distribution coincides with the geometric distribution. The negative binomial distribution often appears in problems related to the randomization of the parameters of a distribution; for example, if $ Y $ is a random variable having, conditionally on $ \lambda $, a Poisson distribution with random parameter $ \lambda $, which in turn has a gamma-distribution with density

$$ \frac{1}{\Gamma ( \mu ) } x ^ {\mu - 1 } e ^ {- \alpha x } ,\ \ x > 0,\ \mu > 0, $$

then the marginal distribution of $ Y $ will be a negative binomial distribution with parameters $ r = \mu $ and $ p = \alpha /( 1+ \alpha ) $. The negative binomial distribution serves as a limiting form of a Pólya distribution.

The sum of independent random variables $ X _ {1} \dots X _ {n} $ which have negative binomial distributions with parameters $ p $ and $ r _ {1} \dots r _ {n} $, respectively, has a negative binomial distribution with parameters $ p $ and $ r _ {1} + \dots + r _ {n} $. For large $ r $ and small $ q $, where $ rq \sim \lambda $, the negative binomial distribution is approximated by the Poisson distribution with parameter $ \lambda $. Many properties of a negative binomial distribution are determined by the fact that it is a generalized Poisson distribution.

References

[1] W. Feller, "An introduction to probability theory and its applications", 1–2, Wiley (1950–1966)

Comments

See also Binomial distribution.

References

[a1] N.L. Johnson, S. Kotz, "Distributions in statistics, discrete distributions" , Wiley (1969)
How to Cite This Entry:
Negative binomial distribution. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Negative_binomial_distribution&oldid=25961
This article was adapted from an original article by A.V. Prokhorov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article